Soundings and Sextants, Part Two, Celestial Navigation Methods

According to Marx’s book on the Spanish flota, ship’s navigators were regarded with scorn and, on many occasions, the denouement to the stranding of a ship’s crew was the assassination or execution of the navigator (71). Up-time celestial navigation methods may thus save not just their reputations, but their lives.

Celestial navigation is the determination of one’s location on earth by observing the apparent position of one or more celestial objects. Celestial navigation is possible because that apparent position is dependent on the location of the observer. It is complicated because the apparent position is also dependent on the rotation of the earth about its axis, and the revolution of the earth about the Sun, and thus on the passage of time. (For the Moon and the planets, it is also affected by their own motion about the Earth and Sun, respectively.)

Most, but not all, of the up-timer’s knowledge of celestial navigation will be gleaned from the encyclopedias. Admiral Simpson, and some of the other navy veterans, will certainly have taken a navigation course at some point during their naval career. And the Ed Board has decreed that there are at least two copies of pre-1995 editions of Bowditch’s American Practical Navigator.

There are three powerboat owners in Grantville. Chances are reasonably good that they are members of the U.S. Power Squadron. The nearest chapter is the Kanawha River Power Squadron in Charleston, West Virginia, and it offers courses in “Seamanship, Piloting, Advanced Piloting and Celestial Navigation 1 and 2.” (wvdnr.gov).

Celestial Latitude and Longitude

The sky looks a bit as though it were painted on the inside of a giant sphere (this is mimicked by a modern planetarium). In fact, Omar Khayyam referred to “that inverted bowl they call the sky.” The stars (including the Sun) appear to move across the “celestial sphere” as a result of the rotation of the Earth, and the revolution of the Earth around the Sun. The apparent motion of the Moon and planets is a composite of their true motion, and the apparent motion generated by the Earth. If we want to use the heavenly bodies as guides to navigation, then we need to be able to describe their positions in the sky.

Celestial coordinates are typically given in one of several different ways, depending on who is using them. First, we have to decide the “origin” (math talk for where we imagine we are standing when we measure the positions). The choices are heliocentric (measured from the center of the Sun), geocentric (from the center of the earth), or topocentric (from a point on the earth’s surface) form. The difference between geocentric and topocentric measurements is likely to be noticeable only for the Moon.

Secondly, we need to choose a coordinate system: the plane from which to measure the angular distance up-or-down, and the direction from which to measure the angular distance left-or-right. There are three principal coordinate systems: equatorial, ecliptic and “alt-az.”

Equatorial Coordinates. At a planetarium, you could project a grid onto the screen, representing the “lines” of celestial latitude and longitude. In the equatorial system, the plane of reference is the celestial equator: the projection, into the sky, of the terrestrial equator. Likewise, the North Celestial Pole (NCP) is directly above the Earth’s North Pole, and the South Celestial Pole below the Earth’s South Pole. Celestial latitude, measured from the equator, is called “declination,” and is measured in degrees, arc-minutes (‘), and arc-seconds (“).

Just as we needed a terrestrial prime meridian from which to measure terrestrial longitude, we need to arbitrarily fix a location for the celestial prime meridian in order to determine right ascension. In 1950, it was defined as passing through the “first point” in the constellation Aries.

There’s no east or west celestial longitude; it is measured either eastward from the first point as 0-24 hours (“right ascension”, RA), or westward as 0-360° (“sidereal hour angle”). RA is stated either like declination, or, because of the relationship of longitude to local time, in hours, minutes (m) and seconds (s) (one minute RA is 15′; one second, 15″).

For any celestial object, there will be a point on the earth’s surface such that the object would be directly overhead. That’s called the “sub-point” (or “geographical point,” GP). As the Earth rotates, etc., the GP will move.

Ecliptic Coordinates. Geocentric equatorial coordinates work well for the Sun and the stars, at least in the short term (years as opposed to centuries), but for the planets, it helps to carry out computations in ecliptic coordinates. The earth’s orbital plane is called the ecliptic, and a line drawn through the center of the Earth, and perpendicular to the ecliptic, defines the North and South Ecliptic Poles. Depending on what you are trying to compute, you can use geocentric or heliocentric coordinates.

Because of precession (the wobbling of the Earth’s axis relative to the plane of the Earth’s orbit), the equatorial coordinates of even the Sun and stars changes slowly with time. (One full precession cycle takes about 26,000 years). Celestial North has to be defined on the basis of the orientation of the Earth’s axis, relative to its orbit, as of a particular time (“epoch”).

Precession causes the NCP to revolve around the North Ecliptic Pole. Thanks to precession, the celestial prime meridian passes through the constellation Pisces.

The Earth’s orbit itself is perturbed by the rest of the Solar System, resulting in changes in the orientation of the major axis and the orbital plane relative to the rest of what I will loosely call “Distant Outer Space.” These changes are just too small and too slow to worry about here.

Horizontal (Alt-Az) Coordinates. A navigator’s observation of a celestial object isn’t likely to be recorded, initially, in equatorial coordinates, but rather in terms of the object’s altitude and azimuth. The altitude is the vertical angle between it and the “celestial horizon,” which in turn is a distant imaginary circle, centered on the observer and level with the observer’s eye, and in a plane perpendicular to the zenith line (from the observer to the point directly overhead, opposite the direction of gravity). The azimuth is its horizontal direction, an angle measured from the direction which points to the North Geographic Pole. The imaginary semicircle running across the sky from north to south is the observer’s meridian.

Alt-Az coordinates are relative to an observer on the earth’s surface, and thus are topocentric. After correcting for observational errors, they can be converted into other coordinates.

Sources of Error in Celestial Navigation

There are several kinds of error which can occur. The first are observational errors, wherein the “read” position, in alt-az coordinates, doesn’t correspond to the actual position of the object at that time. Or there is an error in determining the time at which the observation was made.

Secondly, there can be an error in the prediction of the celestial coordinates. If the navigator is using a published star atlas or catalogue, then this could be an error on the part of whoever computed the published coordinates, or on the part of the navigator, in taking the value from the table, and perhaps in updating it as needed.

Finally, there can be a sight reduction error, that is, an error in the use of the observation and the reference data to compute the latitude and longitude of the ship.

It does no good to worry about computing star positions to the correct milli-arc second if your observational instrument is only accurate to the nearest degree. Hence, in improving the art of navigation, you need to tackle sources of error in their order of importance. Nunez’ Defense of the Sea Chart (1537) said that there was no point in correcting for the meridian of observation, in using solar declination tables, unless the longitude difference was at least six hours, because of the grosser errors resulting from the imprecision with which the astrolabe measured altitude. (Taylor 181).

I am going to ignore sources of error which are always smaller than 1′. Usually those mean an error of about a mile on the ground, but if you are using the “lunar distance” method to measure longitude, a 1′ error in lunar distance corresponds to a 0.5° error (up to 35 miles) in longitude.

Up-time Computer Software

It isn’t likely that there is any navigation software in landlubberly Grantville. However, the high school science department is reported to have at least one astronomy program, and those have data useful in celestial navigation.

There were many amateur astronomy programs available in late 1999/early 2000, but the ones I think most likely to have been acquired for educational purposes are:

Distant Suns 5.1 (3/2000)

Deep Space 5.56 (by 1998)

Dance of the Planets QED edition (1994)

Red Shift 3 (1998)

Starry Nights Deluxe 2.0 (by 1999) or Pro (1/2000)

TheSky v.4 (by early 2000)

Voyager 2 (by 1999)

DeepSky 2000 (1/2000)

Canon says that Johnnie Farrell has a telescope with a “goto.” A certain amount of astronomical data could be extracted, somewhat laboriously, from the “goto.”

Star Data

If we know the locations of the stars in ecliptic or equatorial coordinates, we can use them for navigation. There are three possible sources of this information:

—down-time star catalogs (including atlases and globes), corrected for the passage of time

—up-time star catalogs (books and software), ditto

—post-RoF observations

Down-time Star Data. The most useful compilation is the star catalogue of Tycho Brahe. His “cat D” (1598) provides ecliptic coordinates (nearest 0.5′) for 1004 stars. Tycho was well aware of precession (see below) and, since the catalog was the fruit of years of observation, all star positions were corrected to what they should be for epoch “1601.03”.

Tycho’s accuracy is excellent. Rawlins compared Tycho’s positions to those predicted by combining the Yale Catalog (1982) with “Newcomb’s traditional precession constants” (see “Precession” below). For his 100 “select stars” (the bright stars likely to serve as navigational beacons), the error in either equatorial coordinate was never as much as 6′. The mean error was 1.62′ in RA and 1.48′ in declination.

The greatest weakness of Tycho’s data is that his observatory was in Denmark, and hence his coverage of southern hemisphere stars is poor.

Up-time Star Data. Books for amateur astronomers will explain how to locate stars (and other celestial objects). Ideally, they will specify the location of the star in celestial equatorial coordinates (right ascension and declination) and also state the standard date for which those coordinates were determined.

Pasachoff, Stars and Planets was in the Mannington Middle School Library. The 2000 edition came out in 1999. Appendix 2 gives the calculated mid-1999 equatorial coordinates (to nearest 0.1′) for the 314 brightest stars (down to apparent magnitude 3.55), each identified by its name as given in Bayer’s 1603 atlas. There is also a copy of Burnham’s Celestial Handbook in the high school science department.

The mysterious computer program should have star data specified at least to the nearest arc-minute (as with TellStar, 1985) and more likely to the arc-second (as in Dance of the Planets, 1994).

Precession. The modern star positions, and even those of Tycho’s, are not quite accurate in the 1630s. The discrepancies are primarily the result of precession. Encyclopedia Americana “Equinox” says that the cycle is about 26,000 years, and that precession is at a rate of about 50 arc-seconds/year (which implies a cycle length of 25,920 years). EB11 (“Precession of the Equinoxes”, “Earth”) gives two values for “general precession”, 50.2453 (1850) and 50.2564 (1900) arc-seconds/year.

Both the down-time and up-time star data can be roughly corrected for precession, by any competent down-time astronomer, by assuming that precession occurs at the constant rate suggested by EB11, and then carrying out the appropriate spherical trigonometry calculation. Tycho and Kepler both corrected older data; the value used by Tycho was 51″ (Rawlins 17.)

That astronomy program should be able to precess the modern star positions back to the 1630s and the underlying algorithm is probably more complex (and accurate) than the simple constant precession contemplated by Tycho.

Proper Motion. For some navigational stars (Rigil Kent, Arcturus, Polaris, Zuben-ubi) proper motion (the real motion of the star relative to the solar system) can create a noticeable error (Reis). Obviously, the stars will experience more than ten times as much proper motion in the nearly four centuries separating Pasachoff from the 1630s, as in the three decades elapsed since Tycho’s catalog. The astronomy programs may take proper motion into account.

Recommendations. Assuming, as is likely, that the available software gives star positions with better than the Tychonian arc-minute accuracy, and supports precession to the 1630s, we will probably use the software to read off the correct 163x equatorial coordinates for all the navigational stars. Otherwise, we will probably use Pasachoff’s data for southern hemisphere stars and Tycho’s for the rest, with both adjusted for precession.

In the long-term, astronomers will use telescopes to obtain star positions which are both current and accurate. One of the first catalogs compiled with telescopic assistance, Flamsteed’s, was accurate to 10″ arc (Wakefield 51).

Astronomical and Nautical Almanacs (Ephemerides)

An ephemeris is an almanac which tabulates the positions of an astronomical object as of different times. The difference between the nautical and astronomical almanac is one of emphasis. A modern nautical almanac will list predictions only for the Sun, Moon, and the “navigational” planets, and a stellar reference point, the constellation Ares. It will also identify the locations of the navigational stars (57 nowadays) relative to Ares. An astronomical almanac will cover the other planets and moons, and will provide coordinates for many additional stars. In either case, the solar, lunar and planetary predictions are usually good only for a few years, unless you have a computerized version.

Some internet sources would have you believe that the first nautical almanac was published in 1767. That was merely the first one with “lunar tables” for calculating longitude. The 1545 almanac of Martin Cortes was a long term (1545-1580) almanac, in which the solar declination was calculated by combining values for the month/day, and the year, to obtain the zodiacal position of the sun, and that then used to find the actual declination. In contrast, the almanac of William Bourne (1576) featured a simple look up, but was useable for a much shorter period. A more recent almanac was Davis’s Seaman’s Secrets (1594). It provided a table of the sun’s declination for noon each day for the years 1593-97. (Graham; EB11/Navigation). The down-timers actually could do better than that; Digges’ Prognostications (1553) had a table of the sun’s altitude for every hour of the day at latitude 51.5° N (Taylor 187).

Some of the down-time almanacs also had star data; Mariner’s Mirror (1588) offered the declination and right ascension coordinates for 100 “fixed stars” (Taylor 209).

A modern nautical almanac provides the declination and the Greenwich Hour Angle (GHA), to nearest 0.1′, for each celestial object useful in navigation. The GHA is essentially the angle between the celestial meridian of the object, and the celestial meridian over Greenwich. Values are given for every hour (Greenwich Mean Time, GMT) of every day for the Sun, Moon, and planets, and for every day for the first point of Aries. To get the GHA for a particular star, you add the star’s SHA. (GHA changes 1°/day for the stars, because, thanks to the earth’s orbital movement, the earth doesn’t have to quite complete a full rotation to face the same star a second night.) There is a correction value given, for each day, for the Sun and planets, to allow for interpolation between whole hours. And the Moon’s movement is so irregular that a separate correction value is given for each hour.

The major concern with regard to the down-time manuals is accuracy. For example, for July 23, 1579, when Drake left the California coast, Bourne’s declination tables (1574) were in error by six arc-minutes (Graham). The problem was that Bourne, not knowing Kepler’s laws of planetary motion, had miscalculated the apparent solar movements.

While that was a “model error,” computational errors were common. According to Bowditch, Tables 1 and 2 of Moore’s Practical Navigator (1800) had 3,500 errors. And Astronomer-Royal Maskelyne’s “Requisite Tables” were equally faulty, with 1,024 mistakes in Table 21. (Callaghan 215). The safest course of action is clearly to generate the numbers, and print the manuals, by computer.

Errors can also occur in using tables. Unlike a computer program, a table can’t give celestial positions for every location at any instant of the day. If the observation isn’t for the location and time of day assumed by the table, then for greatest accuracy, you must interpolate between table values. Errors could be made in interpolation, or the seamen could decide not to bother interpolating at all.

Solar declination tables, for example, were calculated as of the local time at a particular location. If the ship were at a different longitude, then its local time was different, and the navigator should make a longitude correction before using the declination, as taught by Hariot. Wright (1599) said that by ignoring longitude, the mariner might be “deceived sometimes 10 or 12 [arc] minutes in taking the sun’s declination. Drake, in circumnavigating the world, ignored the problem. (Graham)

It is worth noting that seamen made calculations using the abacus (Swanick 42) and Gunter’s line, sector and scale (basically devices for graphical solution of trig and log problems).

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Solar Positions. The apparent motion of the Sun is a direct consequence of the real orbital motion of the Earth. There is a systematic error in many seventeenth century predictions of the Sun (and hence of the planets) because of Tycho’s erroneous value (0.018) for the eccentricity of the Earth’s orbit. (Gingerich xix). Cassini (1667) recalculated it as 0.017. Dutton-Smith says that it was 0.01675104 in 1900, and, using his formulae (86), when Grantville popped into the seventeenth century (May 25, 1631, Gregorian), it was 0.016862.

Planetary Positions. The apparent motion of a planet results from the combination of the real motions of that planet and the Earth.The only planets used for navigation are Venus, Mars, Jupiter and Saturn. Their advantage is that they are bright; their disadvantage, it is more difficult to predict their position than that of the “fixed” stars.

Before the up-timers arrived, down-timers predicted planetary appearances using the solar system models of Ptolemy, Copernicus, Tycho, or Kepler.

Kepler, for example, predicted planetary positions through 1637 in the Rudolphine Tables (1627). These predicted planetary positions through 1637. Lorenz Eichstadt (1596-1660) produced sequels in 1634, 1637, and 1639.

Kepler’s Rudolphine Tables competed with the 1632 ephemeris of the Copernican Philip van Lansberge (1561-1632) and the 1622 Astronomia Danica of the “Tychonian” Christen Longomontanus (1562-1647).

Andrea Argoli (1570-1657) based his 1621 ephemeris on pure Copernican theory (adjusted circular heliocentric orbits). In 1634 he published new tables which followed the “Tychonian” model (all planets except the Earth circularly orbit the Sun). Argoli’s predictions for Mars (1650s) were within 10′ arc. His accuracy was less for other planets: Saturn (~40′), Jupiter (30′), Venus (2°), and Mercury (9°). For the “Sun”, it was 8′. (Gingerich xi-xx).

It is perhaps worth mentioning that several later ephemerides authors are alive as of the RoF, including Ismael Boulliau (1605-1694), Noel Durret (1590-1650), Jeremiah Horrocks (1618-1641), and Thomas Streete (1622-1689). They may play a role in post-RoF astronomy.

Down-time mathematician-astronomers are going to learn some very important lessons from the up-timers and their books:

(1) Kepler was on the right track; the planets are, to a first approximation, in elliptical orbits with the Sun at one focus (his first law), they don’t move at a constant velocity (his second law), and the periods are related to the size of the orbits (his third law);

(2) the Keplerian laws aren’t really laws, they are a corollary to a special case of the real law governing planetary motion—Newton’s law of universal gravitation. (Kepler’s laws can be derived if one assumes that there are just two bodies in the universe and one is much more massive than the other.)

(4) there is no exact solution to the N-body problem, but a pragmatic solution is obtainable by approximation methods.

With this information, they can make reasonably accurate short-term predictions of the movements of the planets even without up-time data.

In fact, with just the Keplerian laws, Kepler successfully predicted (in 1627) the transit of Mercury in 1631. The theory of gravitation, in its turn, made it possible for Edmund Halley to recognize that the comet seen in 1682 had previously been observed in 1531 and 1607, and that it would return in 1758.

A planetary orbit is defined by five orbital elements (and a sixth indicates where the planet is in that orbit). Two describe the orbital geometry and the other three the orientation in three dimensions relative to Earth’s orbit. EB11/Planets provides all of the orbital elements for the planets as of 1900/10. Pasachoff (Appendix 9) provides the masses of the Sun and planets, and three of the orbital elements (semimajor axis, eccentricity, and inclination), as of 2000. There are differences; e.g., Mars’ eccentricity is 0.093309 (EB11) or 0.09341 (Pasachoff). Kepler’s value was 0.09265 (Murray 20).

The six elements necessary to specify an elliptical orbit can be determined from three observations. However, there are many different routes of getting from the observational data to the orbital elements, and they vary in terms of accuracy and computation time. Historically, the major post-Newtonian contributions to orbit determination were those of Euler (1744), Lambert (1761-71), Lagrange (1778), Laplace (1780), Olbers (1797), and Gauss (1809)(Dubiago 7-14). The point to note is that some very heavy hitters studied the problem and that it still took over a century to get from Newton’s Principia (1687) to the Gaussian method.

The observations need to be far enough apart so as to “see” the orbit from significantly different perspectives, but close enough together so that the elements haven’t had time to be significantly perturbed. Once the elements are known, you can use additional observations to try to figure out how that orbit is being perturbed. Then you can calculate a more definitive present orbit, and that in turn allows you to later detect smaller or less frequent perturbations.

The basic concept of perturbation is one familiar to seventeenth century mathematicians; the epicycles engrafted on the Ptolemaic (and Copernican) models of the solar system perturbed the basic “circular” orbits of the planet in such a way as to account for discordant observations.

However, perturbation theory is an advanced topic in mathematics and it is by no means clear that the mathematicians of Grantville can teach it. It will need to be rediscovered. Once that happens, what you end up with is that each orbital element, instead of being constant as in the Keplerian model, is a complicated function of time. This is called an “analytical model.”

In theory, if you know the masses, positions and velocities of all significant bodies in the solar system at the same point in time, you can instead use “numerical integration” to determine their positions and velocities in the past and in the future. In essence, you calculate the gravitational forces on each object, and determine how their positions and velocities change over a small time interval. Then you calculate the forces acting over the next time interval, and so on.

This wasn’t feasible until high-speed computers were developed.

In practice, you are combining observational data, of varying reliability, from different dates. So the initial state for the simulation is determined using an analytical model. The simulation is run, generating an ephemeris for a period for which observations exist. The initial state is tweaked until the predicted ephemeris is a “best fit” to the observations. The simulation can be extended to make predictions concerning the past and future perambulations of the bodies.

The accuracy of the predictions will depend on the accuracy of the starting data, and on use of a sufficiently small time step. The smaller the interval, of course, the more computation is necessary.

There were some pre-RoF amateur astronomy programs, such as Dance of the Planets, which had some numerical integration capability.

Among professionals, the trend has been to use numerical integration to generate a “background ephemeris,” and then find the analytical expressions which best fit the data. As of the RoF, the “gold standard” for the planets was the VSOP87 “semi-analytical” model, in which analytical (polynomial and trigonometric) expressions were fitted to the DE200 ephemeris (covering 1600-2169) generated by numerical integration. VSOP87 is believed to be accurate to 0.05 arc second for the modern period and to one arc second over a period of several thousand years. Unfortunately, it also contains thousands of correction terms for each planet (Bretagnon, Table 6). Amateur astronomy programs typically use, at best, a simplified version of VSOP87.

Lunar Positions. The Moon’s orbit about Earth is only approximately elliptical, because of the effect of the Sun, the Earth’s equatorial bulge, the planets, etc. It is thus incredibly difficult to predict.

The average lunar motion is about 30’/hour, but three anomalies (eccentricity, evection and variation) were known to the down-timers. In theory, those are enough to predict the lunar position with an accuracy of about 10′ (Fitzpatrick).

Flamsteed thought that the lunar theory of 1683 was capable of predicting lunar position with an accuracy of at most 12′, and Kollerstrom believes that Newton’s 1702 theory was accurate to 7-8′. The theoretical accuracy would be degraded by computation errors. For example, in 1695-1701, a French almanac had lunar longitude errors in 1695-1701 which sometimes exceeded 30′. (Kollerstrom; cp. Williams 79).

The “gold standard” for the Moon is the “semi-analytical” ELP2000. It is accurate to 1.5″ for 1900-2100 and 20″ for 1500-2500 (Giesen). Amateur astronomy software would probably use a truncated version with arc-minute accuracy.

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The computation of astronomical tables by hand calculations is laborious and error-prone. The process of constructing almanacs would be simplified by amateur astronomy software.

The marketing appeal of these programs often resided in their graphical representations of the sky. But what we want is a program which can print an ephemeris, showing either, for a particular day, the positions of the sun, moon, and planets, or, for a series of days, the positions of a single celestial object.

Secondly, the program must allow the user to specify a date in the 1630s. Starry Night Deluxe, for example, allowed an earliest date of 4713 BC.

Finally, the program must provide accurate results for seventeenth century dates. But we don’t need VSOP87. If the software correctly implements the algorithms published by Meeus’ Astronomical Algorithms (a popular source), it will be accurate for planetary positions in the 1630s to within an arc-second (ProjectPluto.com), which for us is overkill. An NOAA reviewer (Code) said that Voyager II had “planet calculations good to a couple of seconds of arc over 500 years” and implied that it still was not as accurate as Red Shift or Starry Night.

The moon is more of a problem; Meeus only provides about ten arc-second accuracy even for modern times, which, for calculating longitude by lunar distance (see below), will result in longitude errors of about 5′.

If the software becomes unusable (e.g., the disk is copy-protected and becomes unreadable), then we have to compute the tables ourselves. We can use Kepler’s orbital elements, the “up-time” elements, or elements determined by new post-RoF observations. New computer programs (or spreadsheets) can be written, both to derive elements from observations, and to produce ephemerides from the elements.

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For some years after RoF, only Grantville and a few other towns in Europe will have computers and thus those places will produce most ephemerides. The output will probably be to dot matrix printers, which is considered to be a sustainable technology, in order to avoid the transcription errors associated with typesetting.

Navigational Use of the Pole Star

In the northern hemisphere, observation of the Pole Star, Polaris, allows you to determine your latitude, as well as the direction of True North (as distinguished from Compass North). Of course, it is important that you know your constellations. “Columbus’s celestial navigation was almost invariably unfortunate, a litany of wildly wrong latitudes caused by his mistaking other stars for Polaris.” (Phillip-Birt 178)

If Polaris were in fact located at the NCP, then altitude of Polaris would be your latitude, and its bearing would be the direction of True North. Petrus Peregrinus’ Epistola de Magnete (1269) recognized that Polaris moved in a small circle about the north celestial pole. (Polaris is now less than 1° from the NCP. In 1601, Polaris was at 87°08.7′ declination, 5°56.6′ RA—about 3° from NCP (Rawlins 99-S2). Wright (1599) said that the distance was 2d52′ (Graham).)

The Regimento do astrolabio e do quadrante (Lisbon, 1510) evidences that down-timers know how to use the “rule of the north”—based on the orientation of the “guard stars” alpha and beta Ursa Minor—to find the latitude from the altitude of Polaris. If they formed a vertical line, then Polaris was at the same altitude as the celestial north, and no correction was necessary. If they were horizontal, then you had to add or subtract several degrees, depending on whether they pointed west or east. (Taylor 146).

Determining Latitude

Polestar Altitude. I have already alluded to use of the Pole Star to find latitude. That doesn’t work in the daytime, or in the southern hemisphere.

Meridian Sight. The second method requires observing the altitude of a celestial object when it crosses the observer’s meridian. For the sun, this will occur at local noon. (A circumpolar star will cross the meridian twice a day, and the crossings are upper and lower culmination.) Knowing the declination of the sun for the date in question, simple arithmetic yields the latitude.

Of course, that requires both an almanac with a declination table, and the ability to recognize when local noon has arrived. The sun ascends during the morning, and descends during the afternoon. Local noon is the moment at which the solar disk seems to hang motionless in the sky.

Obviously, if you don’t measure the altitude at, precisely, local noon, the computed latitude will be in error. However, at moderate latitudes, around local noon the trajectory of the Sun is fairly flat, i.e., its altitude doesn’t change rapidly. For 40°N, the maximum change of altitude is 0.1′ at 2 min before or after local apparent noon, 0.6′ at 4 min. Sail up to 80°N, and the altitude changes are 0.4′ and 6′, respectively. (Mixter 317)

Double Altitude. Sometimes the weather doesn’t permit a meridian sighting of the sun. If you make two successive observations, and you know the time interval between them, and the sun’s declination (almanac), you can calculate the latitude without bothering to observe the sun at local noon. For this purpose, the watch doesn’t need to keep accurate time over the long term; it just needs to be able to measure a time interval of an hour or two. The measurements should be close to when the object reaches the meridian, and the procedure is more prone to error when the meridian crossing is high in the sky. (Bowditch/1826, 128) A complication is that the first altitude must be corrected for the estimated movement of the ship.

Ex-Meridian Altitude. The weather may be so bad that you can only make one sighting, but close to noon. You can still compute the latitude from the altitude, albeit less reliably. The first item you need is a well-regulated watch; one which keeps good time and which was recently set (perhaps the preceding morning), based on celestial observations, to the local time. The second is an estimated latitude (Bowditch) or longitude. And you need almanac information.

Equal Altitude. This is a special case of the double altitude method. You take a timed morning sighting, and then, in the afternoon, time when the Sun drops back to the same altitude. (You keep the sextant set at the original altitude and let the Sun swim into view.) The time midway between, suitably adjusted, is considered the time of local noon. (Williams 111). Polter (1605) objected to the use of equal altitudes because the declination changes between readings, even though the change is only 1’/hour, at most (Taylor 218).

If the Sun is hidden all day, or poorly located for use of the double altitude method, one may instead observe a star (if there is a visible horizon), a planet, or the Moon, but the latter changes its celestial position rapidly, and this poses computational complications.

Accuracy. Drake’s accuracy (1579) was about 9′ for measurements on shore and 21′ for those at sea (DNG).

Determining Longitude

All methods of determining longitude require comparing local time with the simultaneous time at a reference meridian (e.g., Greenwich). The difference in time, multiplied by 15°, yields the difference in longitude.

No celestial object hovers over a single point of the Earth’s surface (although Polaris comes close). During the course of a day, as a result of the Earth’s rotation, the GP of a star traces a circle on the Earth’s surface. That circle is at a fixed latitude (determined by the declination of the star) but the longitude of the GP can only be determined if you know the local time. And you need to know the longitude of the GP if you want to calculate the longitude of the ship.

Local time. The simplest way, in theory, to know the local time of an observation is to carry it out when the sun has hung in the sky (reached meridian altitude), which is, approximately, local noon. A watch time can be corrected, after the fact, to local time by using the equal altitude method (see “determining latitude” above) to determine the watch time at which local noon occurred. (Preston 172) At sea, it was more common to shoot the Sun when it was bearing east or west and use its altitude, together with computed latitude and estimated longitude, and the Sun’s declination, to calculate the time of observation (Bowditch 155). Star positions can also be used to estimate a local time.

If local noon is determined by a sighting, the time since noon can be tracked by means of an hourglass or, better yet, a simple timepiece. (Even a timepiece that was not suitable for keeping accurate time over the length of a voyage might be reasonably accurate over the hours between a noon-sight and a twilight-sight.) And at night, local time could be determined to perhaps the nearest quarter-hour using a “nocturnal” (see part 1).

Reference time. The reference time may be determined either by observing some celestial event (which happens essentially simultaneously for both the reference observatory and the ship’s location), or by inspection of a chronometer set previously to the reference and which has kept “consistent” time (it loses or gains time in a predictable manner) since then.

The celestial events which have been used for longitude determinations include jovian moon eclipses, lunar eclipses, lunar occultations, and particular angular separations of the moon from the Sun or stars.

Jovian Moon Method. In theory, a reference time could be determined by noting when the moons of Jupiter passed into or out of its shadow, and comparing it to the times stated in an ephemeris computed for a location of known longitude. When a predicted immersion or emersion was observed, a clock was set to the ephemeris time. The next day, the observer noted the clock time at which the sun peaked (local noon). You then calculated the longitude, hoping that in the course of a day, the clock wouldn’t lose or gain too much time from the true reference time.

The ephemeris for Paris was calculated by Cassini in 1668 and by 1696 Cassini published a map of the world which used longitudes determined by this method. Unfortunately, the method was impractical on shipboard. The necessary telescope (15-20 feet long) had a narrow field of view, so it would be difficult to keep the moons under observation while the ship pitched and rolled. If you were using a pendulum clock, then there was the further problem that the clock wouldn’t work properly, even over the relatively short time interval between the two necessary observations The experienced astronomer Halley tried, but concluded that the Jovian eclipses were “absolutely unfit at sea” (Mentzer; Wakefield 86-7).

Just as well. Cassini’s tables were in substantial error because he failed to consider the effect of the finite speed of light (discovered by Roemer, 1676) on the time of observation of Jovian eclipses (Wakefield 164).

Lunar eclipse. A lunar eclipse occurs when the Moon passes through the Earth’s shadow. It is observable from anywhere on the night hemisphere, and begins and ends at the same time for all observers. If you have an almanac giving the time the eclipse begins or ends for a reference site of known longitude, you can compare that reference time to the local time. Unfortunately, lunar eclipses occur only a few times a year, are difficult to time, and in practice yield an accuracy of only perhaps 0.5-1.5° (Espenak; Oliver).

Lunar occultation. The moon takes about 29.5 days (its synodic period) to travel 360° in celestial longitude, so its change in celestial longitude is about half a degree per hour. In contrast, the stars have essentially fixed celestial longitudes. Hence, the movement of the moon, relative to the stars, could be used to judge the reference time.

Initially, it was proposed that astronomers predict the times that the moon would “occult” (pass in front of) various stars. Unfortunately, while we think of the moon as large, its angular size is half a degree—that of a penny held 2.29 meters away. On a ship at sea, you are normally going to be able to identify the brighter stars, and the odds are not great that, on a particular night, one of these will be occulted by the moon.

Lunar Distance (“Lunars”). Hence, astronomers instead predicted the “lunar distance,” the angular distance between the moon and a celestial reference point (a star or the Sun), for different hours of the day, day after day. (Although the Sun moves against the sky, its celestial longitude can be predicted with accuracy.)

The lunar distance method was proposed by Regiomontanus (1475) and Werner (1514), but it wasn’t seriously pursued until the 1700s. There are several methods of calculating the “lunar distance”, with different tradeoffs between accuracy and speed; Bowditch’s 1802 method was the first one considered practical by mariners. While we expect to find copies of “Bowditch” in Grantville, it is unlikely that we will find an edition which contains either the original lunar distance methods (dropped in 1880) or the replacement method (dropped from the Appendix in 1914).

Although the rapid movement of the Moon makes it a potentially usable celestial clock, it was a somewhat frustrating one in practice. All errors in observation, prediction and computation are multiplied by the ratio of the earth’s rate of rotation (15°/hour) to the rate of change in lunar longitude (~0.5°/hour): 29.5.

Lunars were difficult from an observational standpoint. Normally, a sextant is held vertically, and used to measure altitude above the horizon. For lunars, it had to be held obliquely, depending on where the Moon and the reference object were located.

The observed angle would be affected, like any other sighting, by dip and refraction. However, to correct the lunar distance, you needed to know the altitudes of the Moon and its “partner.” So that meant, ideally, taking three simultaneous sextant measurements: the two altitudes, and the lunar distance. That was usually impractical, so what was done instead was to 1) measure the altitudes of both objects, 2) then the lunar distance, and 3) the altitudes again. The “before” and “after” altitudes for each object were averaged together to estimate the altitudes at the time of the lunar distance measurement.

Then you had to apply the tables. Their accuracy depended on the astronomers’ understanding of lunar movements. Predicting lunar position is complicated because the moon’s orbit is strongly perturbed by the Sun, so it can’t be calculated purely by Keplerian methods. In the 1783 Nautical Almanac, the average error was 14″ in ecliptic latitude and 30″ in longitude. In 1817, the average latitude and longitude errors were 5″. (Williams 96)

Then the ship’s longitude had to be computed correctly. In practice, “longitude by chronometer” (see below) was perhaps ten times more accurate than “longitude by lunar distance,” because the lunar observations and calculations were so complex and prone to error (Sobel 162). Preston (180) says that in the early nineteenth century, it was not unusual for lunars to yield a 30′ error in the longitude. Lewis and Clark used the lunar distance method, and their errors in longitude were as great as 185′ for moon-star and 76′ for moon-sun measurements (185).

Chronometers. In 1530, Gemma Frisius pointed out that if one had a good clock, one could set it according to the time at a location of known longitude. Multiplying the time difference in hours between the reference clock time when the sun peaked (local noon), and noon, by 15 then gave the longitude difference in degrees from the known longitude.

An alternative to reading the chronometer time at local noon is to take two readings, one before noon and one after, when the sun is at the same altitude. The time of local noon is then the average of the two equal altitude times (Schlereth 96; Preston 172).

Practical use of this method had to await the development of a ship-friendly clock.

The down-time clocks have a cumulative error of 10-15 minutes/day. “After a few weeks at sea, clock error could correspond to a longitude error as wide as the ocean.” (William 78; Mixter, 271; Wakefield 136).

The first reliable marine chronometer was designed and built by Harrison in the late eighteenth century. His H-4 (1760), lost only five seconds after 81 days at sea. After another two months, its temperature-adjusted total error was still under two minutes (Sobel 120-1).

The Harrison chronometer’s principal weakness was the time and expense necessary to build it. The copy of H-4 made by Kendall (1770) cost 500 pounds, and took two years to construct. Kendall told the Board of Longitude, “I am of the opinion that it would be many years (if ever) before a watch of the same kind . . . could be afforded for 200 pounds.” (200 pounds in 1776 was equivalent in buying power to 150 pounds in 1632; enough to buy a large yacht of 25-35 tons burden.) In contrast, a sextant and a set of lunar distance tables would cost a mere 20 pounds (153).

Later clockmakers nonetheless brought costs down by having the less critical parts made by lesser craftsmen. By the 1780s, you could buy an Arnold box chronometer for 80 pounds, or an Earnshaw for 65. There was also the Arnold pocket chronometer, which only gained or lost three seconds a day. (Sobel 156-63)

The 1632 Tech Dead Horses page comments, “Yes, we can make a lot of $ using up-time clocks, and no, no one is going to recreate Harrison’s clock, it’s way simpler to copy something newer. And people will.” And that’s where I am going to leave it. This is an article on navigation, not horology.

It is not necessary that the clock keep perfect time. What is necessary is that its error, if any, be known and predictable. A modern chronometer is set, in port, to approximate GMT, and certified as to its initial departure from GMT, the average rate at which it gains or loses time, and the date of the time check. (Mixter 272)

Most of my sources emphasized how much more difficult it was to determine longitude at sea than on land. However, maritime travel did have the advantage of being fast; an Atlantic crossing was something like two months by sail, less by steam. Hence, the chronometer’s time error—especially the unpredictable error—wouldn’t have the chance to accumulate to unbearable levels before you reached a port of known longitude, and could re-calibrate it. However, the Lewis and Clark expedition (1803-6) took years, and its rigors, in some respects, were greater than those of a sea passage. Consequently, L&C periodically re-calibrated their chronometer by the lunar distance method. This revealed considerable rate variation; it lost 15.6 sec/day in the summer 1803, 36 in late November, and 46.44 in mid-December. (Huxtable 1.4)

“Lunars” were only rarely used after 1850, in view of the convenience and accuracy of calculations based on improved chronometers. One use was to verify that the chronometers were still in working order. The Nautical Almanac stopped publishing lunar distances in 1907.

Modern Celestial Cross-Fix

Lines of Position. Earlier celestial navigation methods required making an observation at a special time (e.g., local noon) which simplified the calculations. Latitude and longitude were determined separately. But you can determine both ship’s latitude and longitude simultaneously if you know the reference time (per chronometer) and take two (at most three) sextant readings (“time sights”) on a celestial object whose position is tabulated in a nautical almanac.

From the almanac and the reference time, you know the GP of the celestial object. The sextant reading defines a circle about the GP, whose radius equals 90°-altitude. If a second mariner simultaneously takes a sextant reading of a second object, you get a second circle, and the ship must be at one of the two intersection points. Usually your dead reckoning from the last known position will tell you which of the two is right, but if it doesn’t you can take a reading of a third object to be sure.

If the sextant readings aren’t simultaneous, the older reading is “advanced” (moved, based on ship’s course and speed) so that they are effectively simultaneous. This is called a “running fix.”

It is inconvenient to plot these circles, which are usually very large, on a globe or a chart. Fortunately, a small enough arc of a large circle can be approximated by a straight line, and the position is then where the two “lines of position” (LOPs, Sumner lines) cross. These can be plotted on a large-scale map (EB11/Navigation).

Graphic methods work best when the two LOPs meet at a substantial angle, and the objects can be picked to ensure this. Sun and moon will generate LOPs crossing each other at an angle of 45° or better perhaps ten days a month (Schlereth 77). Or you can pick stars which are in different quadrants of the sky.

As described in EB11, Saint Hilaire suggested (1875) a major improvement in the 1847 Sumner method. This involved using an assumed position to compute an expected altitude and azimuth for the GP, then plotting the LOP perpendicular to the computed azimuth line, moving the LOP toward or away from the assumed position to account for the difference between the expected altitude and the observed (after correction for observational errors) altitude.

The expected altitude and azimuth can be obtained by spherical trigonometry, by solving the navigational triangle (two sightings yields two linked triangles). This is formed by the assumed position (AP), the GP of a sighted celestial object, and the nearest pole (P). For each triangle, we have three sides, whose lengths are:

GP-AP: 90-expected altitude

GP-P: 90-object declination (looked up in almanac)

AP-P: 90-assumed latitude

The angle with vertex at P is the expected difference in longitude (meridian angle) between AP (assumed longitude) and GP (looked up). The angle with vertex at AP is the expected azimuth (or 360°-azimuth) of the GP, if the ship is at the assumed position. The sides AP-P and GP-P, and the meridian angle, are used to calculate the expected altitude and azimuth.

At local noon, the meridian angle becomes zero and the triangle degenerates into a straight line, vastly simplifying computations.

Log and Trig tables. Sight reduction (the conversion of observations to positions) is heavily dependent on knowledge of spherical geometry and trigonometry. Many different formulae were known by the early seventeenth century. Typically, these required multiplication of trigonometric functions. In the sixteenth century, the trigonometric functions of angles had been calculated to fifteen decimal places. Logarithms were important because instead of multiplying trigonometric functions you could just add their logarithms. Napier published logarithmic tables in 1614, and by 1624 they had been computed to fourteen decimal places (Williams 47-54).

Sight Reduction Tables. So sailors don’t have to do trigonometric calculations, sight reduction tables have been prepared. They compile precomputed navigation triangles, typically covering each possible whole degree value of the meridian angle, latitude, and declination. Unlike almanacs, these are always valid; the math doesn’t change.

To use the tables, you assume a ship position which is at a whole degree latitude and longitude within 30′ of the dead reckoning position, and calculate the meridian angle from the longitude and the almanac listing of the “hour angle” of the object. The declination is in the almanac. Together with the assumed latitude, you use the SRTs to find the expected altitude (nearest 0.1′) and azimuth (0.1°) of the object. You then compute the altitude difference and draw the LOP accordingly.

There is no assurance that any sight reduction tables will exist in Grantville. If anyone has them, it is Jesse Wood, since they are also used in aerial navigation. If they don’t exist, they can be generated by computer, and printed in volume on dot matrix printers. In fact, that would probably be better than trying to typeset Jesse’s copy, since it would avoid typesetting errors.

Conclusion

Jack London, sailing in the Pacific, noted that every step of the navigational art must be performed correctly, or else the captain will hear, ” ‘Breakers ahead!’ some pleasant night, receive a nice sea-bath, and be given the delightful diversion of fighting the way to shore through a horde of man-eating sharks.” The up-timer’s mathematical knowledge and mechanical skills will make navigation a bit safer than it was before the Ring of Fire.

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Author’s Note

The bibliography is included in the Navigation Addendum posted to www.1632.org.